whereThe need of mass renormalizations in the case of massive quarks is understood.
we have suppressed the dependence of the quantities on external kinematics and masses
as well as on auxiliary dimensional scales except the dimensional regulator \(\epsilon \) ,
(for a detailed exposition, see e.g. [1]}, [2]}, [3]}, [4]}
and references therein).
The bare amplitude \(\hat{\mathcal {A}}(\epsilon )\)  and the finite pole-subtracted amplitude
\(\hat{\mathcal {F}}(\epsilon )\)  should be viewed as vectors in the color space of the external particles,
and the multiplicative singular IR-factor \(\hat{\mathcal {Z}}_{\mathrm {IR}}(\epsilon )\)  as a matrix.
The RS-dependent singular factors \(\mathcal {Z}_{\mathrm {UV}}(\epsilon )\)  and \(\hat{\mathcal {Z}}_{\mathrm {IR}}(\epsilon )\) 
encode all UV and IR pole-singularities of \(\hat{\mathcal {A}}(\epsilon )\) , and are independent
of the detailed kinematic configuration, such as polarization states, of the external resolved particles.
(This is the meaning of “factorization”.)
By the very meaning of pole factorization in eq. (REF ),
\(\hat{\mathcal {F}}(\epsilon )\)  is regular in \(\epsilon \)  and has a finite 4-dimensional limit,
\(\hat{\mathcal {F}}(\epsilon = 0)\) .
We call this quantity the (4-dimensional) finite remainder of \(\hat{\mathcal {A}}(\epsilon )\) 
defined by subtracting all poles minimally by the multiplicative factors
as sketched in eq. (REF ).
